34
2 Geometry of the State Space of Quantum Mechanics
where σ f is a vector field defined on the whole P(H) by
x (σ f (x), v) := −d x f (v), ∀v ∈ T x P(H).
(2.3.28)
The equation (2.3.27) has the form of evolution equation of classical mechanics
in terms of Poisson brackets.
2.3.11 Let us add a note concerning possible generalizations of the here presented
dynamics. Since P(H) is a symplectic manifold, more general Hamiltonian evolutions can be defined on it than the evolutions corresponding to linear Schrödinger
equations (2.3.22). We can choose instead of the function f A : P(H) → R as a (‘classical’) Hamiltonian an arbitrary ‘sufficiently differentiable’ function h : P(H) → R.
Then we obtain from the corresponding Hamiltonian dynamics on the infinite dimensional symplectic manifold P(H) evolution of QM-vector states in H, which cannot
be described (in general) by a linear Schrödinger equation. This situation is described
in many details in [37].
2 Geometry of the State Space of Quantum Mechanics
where σ f is a vector field defined on the whole P(H) by
x (σ f (x), v) := −d x f (v), ∀v ∈ T x P(H).
(2.3.28)
The equation (2.3.27) has the form of evolution equation of classical mechanics
in terms of Poisson brackets.
2.3.11 Let us add a note concerning possible generalizations of the here presented
dynamics. Since P(H) is a symplectic manifold, more general Hamiltonian evolutions can be defined on it than the evolutions corresponding to linear Schrödinger
equations (2.3.22). We can choose instead of the function f A : P(H) → R as a (‘classical’) Hamiltonian an arbitrary ‘sufficiently differentiable’ function h : P(H) → R.
Then we obtain from the corresponding Hamiltonian dynamics on the infinite dimensional symplectic manifold P(H) evolution of QM-vector states in H, which cannot
be described (in general) by a linear Schrödinger equation. This situation is described
in many details in [37].